Systems and methods for fully coupled models for crowd navigation

ABSTRACT

Systems and methods for utilizing interactive Gaussian processes for crowd navigation are provided. In one embodiment, a system for a crowd navigation includes a processor, a statistical module, and a model module. The processor receives sensor data. The statistical module identifies a number of agents in a physical environment based on the sensor data. The statistical module further calculates a set of Gaussian processes. The set of Gaussian processes includes a Gaussian Process for each agent of the number of agents. The statistical module further determines an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents. The model module generates a model of the number of agents by applying the objective function to the set of Gaussian processes. The model includes a convex configuration of the number of agents in the physical environment.

CROSS-REFERENCE TO RELATED APPLICATION

This application is related to U.S. Provisional Application Ser. No. 62/937,334 filed on Nov. 19, 2019, which is expressly incorporated herein by reference. Additionally, this application is related to U.S. Provisional Application Ser. No. 62/799,481 filed on Jan. 31, 2019, which is expressly incorporated herein by reference. Furthermore, the application is related to U.S. Provisional Application Ser. No. 62/899,676 filed on Sep. 12, 2019, which is expressly incorporated herein by reference. This application is also related to U.S. Non-provisional application Ser. No. 16/743,777 filed on Jan. 15, 2020, which is expressly incorporated herein by reference.

BACKGROUND

Constructing realistic, real time, human-robot interaction models is a core challenge in crowd navigation. Typically, work has focused on robot-agent coupling, referred to as a first order interaction, while ignoring agent-agent coupling, referred to as a second order interaction. In 2010, a freezing robot problem (FRP) was discovered when independently modeling a host for crowd navigation. In particular, the host navigating a group of humans caused the host to freeze (or take unnecessary evasive maneuvers that negatively impacted efficiency) as congestion worsened. The FRP was demonstrated experimentally in a six month university cafeteria study by showing that the host-agent independence assumption caused a 3× decrement in complete stopping for densities above 0.55 people/m². Further, the FRP had been reproduced in multiple studies indicating its impact on crowd navigation. Although modeling of the host in a crowd has been attempted, very little is known about what form a collision avoidance function should take. Further, merely formulating the collision avoidance function is insufficient because inference is non-trivial.

BRIEF DESCRIPTION

According to one embodiment, a system for a crowd navigation of a host vehicle among a plurality of agents is provided. The system includes a processor, a statistical module, and a model module. The processor receives sensor data. The statistical module identifies a number of agents in a physical environment based on the sensor data. The statistical module further calculates a set of Gaussian processes. The set of Gaussian processes includes a Gaussian Process for each agent of the number of agents. The statistical module further determines an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents. The model module generates a model of the number of agents by applying the objective function to the set of Gaussian processes. The model includes a convex configuration of the number of agents in the physical environment.

According to another embodiment, a method for crowd navigation of a host vehicle among a plurality of agents is provided. The method includes identifying a number of agents in a physical environment based on the sensor data. The method also includes calculating a set of Gaussian processes. The set of Gaussian processes includes a Gaussian Process for each agent of the number of agents. The method further includes determining an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents. The method yet further includes generating a model of the number of agents by applying the objective function to the set of Gaussian processes. The model includes a convex configuration of the number of agents in the physical environment.

According to yet another embodiment, a non-transitory computer readable storage medium storing instructions that, when executed by a computer having a processor, cause the computer to perform a method for crowd navigation of a host vehicle among a plurality of agents. The method includes identifying a number of agents in a physical environment based on the sensor data. The method also includes calculating a set of Gaussian processes. The set of Gaussian processes includes a Gaussian Process for each agent of the number of agents. The method further includes determining an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents. The method yet further includes generating a model of the number of agents by applying the objective function to the set of Gaussian processes. The model includes a convex configuration.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an exemplary component diagram of a system for crowd navigation, according to one aspect.

FIG. 2 is an exemplary process flow of a method for crowd navigation, according to one aspect.

FIG. 3A is another exemplary agent environment for a system for crowd navigation, according to one aspect.

FIG. 3B is another exemplary agent environment for a system for crowd navigation, according to one aspect.

FIG. 3C is another exemplary agent environment for a system for crowd navigation, according to one aspect.

FIG. 4 is an illustration of an example computer-readable medium or computer-readable device including processor-executable instructions configured to embody one or more of the provisions set forth herein, according to one aspect.

FIG. 5 is an illustration of an example computing environment where one or more of the provisions set forth herein are implemented, according to one aspect.

DETAILED DESCRIPTION

Systems and methods for crowd navigation utilizing fully coupled models of first order interactions and second order interactions to mitigate the FRP and unnecessary evasive maneuvers that negatively impact safety and efficiency. Considering both the first order interactions and the second order interactions substantially improves safety, efficiency, and performance.

In particular, the systems and methods here in provide an inference on a statistically valid joint over Gaussian process mixture (GPM) agent models by first formulating the joint using a generic interaction function and then leverage the GP mixtures to constrain the joint collision avoidance function.

The means are interpreted as optimization variables which are modulated by the GP covariances. Effectively, then, we are optimizing over a function space—rather than a state space of time ordered positions—because the covariance encodes a smooth evolution of trajectories as they mutually shape each other. Further, by using initialization functions, the real time approach is competitive with exhaustive search of the non-convex solution space.

A system and method for crowd navigation utilizing interacting Gaussian mixture models such as zero free-parameter Gaussian processes (zpIGP) for crowd navigation in congested environments is disclosed. In particular, the agent models may be Gaussian mixture models (GMM). For example, dynamical models with Gaussian noise are GMMs; deep networks with LSTM outputs. Here, a joint collision avoidance function composed of GMM/GPM agent models is provided, thereby providing insight into the optimality structure of this important class of crowd navigation distributions. Moreover, the joint collision avoidance function provides an efficiency statistical profile competitive with humans that outperforms the other crowd navigation systems and methods.

Definitions

The following includes definitions of selected terms employed herein. The definitions include various examples and/or forms of components that fall within the scope of a term and that can be used for implementation. The examples are not intended to be limiting. Furthermore, the components discussed herein, can be combined, omitted, or organized with other components or into different architectures.

“Bus,” as used herein, refers to an interconnected architecture that is operably connected to other computer components inside a computer or between computers. The bus can transfer data between the computer components. The bus can be a memory bus, a memory processor, a peripheral bus, an external bus, a crossbar switch, and/or a local bus, among others. The bus can also be a vehicle bus that interconnects components inside a vehicle using protocols such as Media Oriented Systems Transport (MOST), Controller Area network (CAN), Local Interconnect network (LIN), among others.

“Component,” as used herein, refers to a computer-related entity (e.g., hardware, firmware, instructions in execution, combinations thereof). Computer components may include, for example, a process running on a processor, a processor, an object, an executable, a thread of execution, and a computer. A computer component(s) can reside within a process and/or thread. A computer component can be localized on one computer and/or can be distributed between multiple computers.

“Computer communication,” as used herein, refers to a communication between two or more communicating devices (e.g., computer, personal digital assistant, cellular telephone, network device, vehicle, vehicle computing device, infrastructure device, roadside equipment) and can be, for example, a network transfer, a data transfer, a file transfer, an applet transfer, an email, a hypertext transfer protocol (HTTP) transfer, and so on. A computer communication can occur across any type of wired or wireless system and/or network having any type of configuration, for example, a local area network (LAN), a personal area network (PAN), a wireless personal area network (WPAN), a wireless network (WAN), a wide area network (WAN), a metropolitan area network (MAN), a virtual private network (VPN), a cellular network, a token ring network, a point-to-point network, an ad hoc network, a mobile ad hoc network, a vehicular ad hoc network (VANET), a vehicle-to-vehicle (V2V) network, a vehicle-to-everything (V2X) network, a vehicle-to-infrastructure (V2I) network, among others. Computer communication can utilize any type of wired, wireless, or network communication protocol including, but not limited to, Ethernet (e.g., IEEE 802.3), WiFi (e.g., IEEE 802.11), communications access for land mobiles (CALM), WiMax, Bluetooth, Zigbee, ultra-wideband (UWAB), multiple-input and multiple-output (MIMO), telecommunications and/or cellular network communication (e.g., SMS, MMS, 3G, 4G, LTE, 5G, GSM, CDMA, WAVE), satellite, dedicated short range communication (DSRC), among others.

“Communication interface” as used herein can include input and/or output devices for receiving input and/or devices for outputting data. The input and/or output can be for controlling different vehicle features, which include various vehicle components, systems, and subsystems. Specifically, the term “input device” includes, but is not limited to: keyboard, microphones, pointing and selection devices, cameras, imaging devices, video cards, displays, push buttons, rotary knobs, and the like. The term “input device” additionally includes graphical input controls that take place within a user interface which can be displayed by various types of mechanisms such as software and hardware-based controls, interfaces, touch screens, touch pads or plug and play devices. An “output device” includes, but is not limited to, display devices, and other devices for outputting information and functions.

“Computer-readable medium,” as used herein, refers to a non-transitory medium that stores instructions and/or data. A computer-readable medium can take forms, including, but not limited to, non-volatile media, and volatile media. Non-volatile media can include, for example, optical disks, magnetic disks, and so on. Volatile media can include, for example, semiconductor memories, dynamic memory, and so on. Common forms of a computer-readable medium can include, but are not limited to, a floppy disk, a flexible disk, a hard disk, a magnetic tape, other magnetic medium, an ASIC, a CD, other optical medium, a RAM, a ROM, a memory chip or card, a memory stick, and other media from which a computer, a processor or other electronic device can read.

“Database,” as used herein, is used to refer to a table. In other examples, “database” can be used to refer to a set of tables. In still other examples, “database” can refer to a set of data stores and methods for accessing and/or manipulating those data stores. In one embodiment, a database can be stored, for example, at a disk, data store, and/or a memory. A database may be stored locally or remotely and accessed via a network.

“Data store,” as used herein can be, for example, a magnetic disk drive, a solid-state disk drive, a floppy disk drive, a tape drive, a Zip drive, a flash memory card, and/or a memory stick. Furthermore, the disk can be a CD-ROM (compact disk ROM), a CD recordable drive (CD-R drive), a CD rewritable drive (CD-RW drive), and/or a digital video ROM drive (DVD ROM). The disk can store an operating system that controls or allocates resources of a computing device.

“Display,” as used herein can include, but is not limited to, LED display panels, LCD display panels, CRT display, touch screen displays, among others, that often display information. The display can receive input (e.g., touch input, keyboard input, input from various other input devices, etc.) from a user. The display can be accessible through various devices, for example, though a remote system. The display may also be physically located on a portable device, mobility device, or host.

“Logic circuitry,” as used herein, includes, but is not limited to, hardware, firmware, a non-transitory computer readable medium that stores instructions, instructions in execution on a machine, and/or to cause (e.g., execute) an action(s) from another logic circuitry, module, method and/or system. Logic circuitry can include and/or be a part of a processor controlled by an algorithm, a discrete logic (e.g., ASIC), an analog circuit, a digital circuit, a programmed logic device, a memory device containing instructions, and so on. Logic can include one or more gates, combinations of gates, or other circuit components. Where multiple logics are described, it can be possible to incorporate the multiple logics into one physical logic. Similarly, where a single logic is described, it can be possible to distribute that single logic between multiple physical logics.

“Memory,” as used herein can include volatile memory and/or nonvolatile memory. Non-volatile memory can include, for example, ROM (read only memory), PROM (programmable read only memory), EPROM (erasable PROM), and EEPROM (electrically erasable PROM). Volatile memory can include, for example, RAM (random access memory), synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDRSDRAM), and direct RAM bus RAM (DRRAM). The memory can store an operating system that controls or allocates resources of a computing device.

“Module,” as used herein, includes, but is not limited to, non-transitory computer readable medium that stores instructions, instructions in execution on a machine, hardware, firmware, software in execution on a machine, and/or combinations of each to perform a function(s) or an action(s), and/or to cause a function or action from another module, method, and/or system. A module can also include logic, a software-controlled microprocessor, a discrete logic circuit, an analog circuit, a digital circuit, a programmed logic device, a memory device containing executing instructions, logic gates, a combination of gates, and/or other circuit components. Multiple modules can be combined into one module and single modules can be distributed among multiple modules.

“Operable connection,” or a connection by which entities are “operably connected,” is one in which signals, physical communications, and/or logical communications can be sent and/or received. An operable connection can include a wireless interface, firmware interface, a physical interface, a data interface, and/or an electrical interface.

“Portable device,” as used herein, is a computing device typically having a display screen with user input (e.g., touch, keyboard) and a processor for computing. Portable devices include, but are not limited to, handheld devices, mobile devices, smart phones, laptops, tablets, e-readers, smart speakers. In some embodiments, a “portable device” could refer to a remote device that includes a processor for computing and/or a communication interface for receiving and transmitting data remotely.

“Processor,” as used herein, processes signals and performs general computing and arithmetic functions. Signals processed by the processor can include digital signals, data signals, computer instructions, processor instructions, messages, a bit, a bit stream, that can be received, transmitted and/or detected. Generally, the processor can be a variety of various processors including multiple single and multicore processors and co-processors and other multiple single and multicore processor and co-processor architectures. The processor can include logic circuitry to execute actions and/or algorithms.

“Vehicle,” as used herein, refers to any moving vehicle that is capable of carrying one or more users and is powered by any form of energy. The term “vehicle” includes, but is not limited to cars, trucks, vans, minivans, SUVs, motorcycles, scooters, boats, go-karts, amusement ride cars, rail transport, personal watercraft, and aircraft. In some cases, a motor vehicle includes one or more engines. Further, the term “vehicle” can refer to an electric vehicle (EV) that is capable of carrying one or more users and is powered entirely or partially by one or more electric motors powered by an electric battery. The EV can include battery electric vehicles (BEV) and plug-in hybrid electric vehicles (PHEV). The term “vehicle” can also refer to an autonomous vehicle and/or self-driving vehicle powered by any form of energy. The autonomous vehicle can carry one or more users. Further, the term “vehicle” can include vehicles that are automated or non-automated with pre-determined paths or free-moving vehicles.

“Vehicle system,” as used herein can include, but is not limited to, any automatic or manual systems that can be used to enhance the vehicle, driving, and/or safety. Exemplary vehicle systems include, but are not limited to: an electronic stability control system, an anti-lock brake system, a brake assist system, an automatic brake prefill system, a low speed follow system, a cruise control system, a collision warning system, a collision mitigation braking system, an auto cruise control system, a lane departure warning system, a blind spot indicator system, a lane keep assist system, a navigation system, a steering system, a transmission system, brake pedal systems, an electronic power steering system, visual devices (e.g., camera systems, proximity sensor systems), a climate control system, an electronic pretensioning system, a monitoring system, a passenger detection system, a vehicle suspension system, a vehicle seat configuration system, a vehicle cabin lighting system, an audio system, a sensory system, an interior or exterior camera system among others.

I. System Overview

Referring now to the drawings, the drawings are for purposes of illustrating one or more exemplary embodiments and not for purposes of limiting the same. FIG. 1 is an exemplary component diagram of an operating environment 100 for utilizing fully coupled Gaussian mixture models for crowd navigation, according to one aspect. The operating environment 100 includes a sensor module 102, a computing device 104, and operational systems 106 interconnected by a bus 108. The components of the operating environment 100, as well as the components of other systems, hardware architectures, and software architectures discussed herein, may be combined, omitted, or organized into different architectures for various embodiments. The computing device 104 may be implemented with a device or remotely stored.

Considering a vehicle embodiment, the computing device 104 may be implemented as part of a telematics unit, a head unit, a navigation unit, an infotainment unit, an electronic control unit, among others of a host, such as the host 302 shown in FIG. 3. The host 302 may be a vehicle, robot, or other self-propelled machine. In other embodiments, the components and functions of the computing device 104 can be implemented, for example, with other devices 530 (e.g., a portable device) or another device connected via a network (e.g., a network 132). The computing device 104 may be capable of providing wired or wireless computer communications utilizing various protocols to send/receive electronic signals internally to/from components of the operating environment 100. Additionally, the computing device 104 may be operably connected for internal computer communication via the bus 108 (e.g., a Controller Area Network (CAN) or a Local Interconnect Network (LIN) protocol bus) to facilitate data input and output between the computing device 104 and the components of the operating environment 100.

The computing device 104 includes a processor 112, a memory 114, a data store 116, and a communication interface 118, which are each operably connected for computer communication via a bus 108 and/or other wired and wireless technologies. The communication interface 118 provides software and hardware to facilitate data input and output between the components of the computing device 104 and other components, networks, and data sources, which will be described herein. Additionally, the computing device 104 also includes a statistical module 120 and a model module 122, for crowd navigation facilitated by the components of the operating environment 100.

The statistical module 120 may be an artificial neural network that acts as a framework for machine learning, including deep learning. The model module 122 may be a decoder that converts the data generated by the statistical module 120 to a model of the physical environments shown in FIGS. 3A, 3B, and 3C. The predicted labels of the model module 122 may be labels that correspond to future actions based on the sensor data 110. Continuing the vehicular example given above, the label may correspond to a predicted maneuver of the host 302. In some embodiments, the predicted maneuver may include a series of maneuvers (e.g., going-straight, right-turn, left-turn, decelerate, etc.). In some embodiments, the labels may be directed to maneuvers of the vehicle. Additionally, or alternatively the labels may be learned or received from a remote server (not shown).

The computing device 104 is also operably connected for computer communication (e.g., via the bus 108 and/or the communications interface 118) to one or more operational systems 106. The operational systems 106 can include, but are not limited to, any automatic or manual systems that can be used to enhance the device, operation, and/or safety. The operational systems 106 may dependent on the implementation. For example, given a vehicular embodiment, the operational systems 106 include a brake system 124, a path planning module 126, a notification system 128, and a sensor system 130 according to an exemplary embodiment. The brake system 124 monitors, analyses, and calculates braking information and facilitates features like anti-lock brake system, a brake assist system, and an automatic brake prefill system. The path planning module 126 monitors, analyses, operates the device to some degree. For example, the path planning module 126 may store, calculate, and provide directional information and facilitates features like vectoring and obstacle avoidance among others. The notification system 128 identifies notifications, generates notifications, and facilitates communication.

The operational systems 106 also include and/or are operably connected for computer communication to the sensor system 130. The sensor system 130 provides and/or senses information associated with a device (e.g., the host 302), the operating environment 100, an environment of the device, and/or the operational systems 106. The sensor system 130 can include, but is not limited to, environmental sensors, vehicle speed sensors, accelerator pedal sensors, brake sensors, wheel sensors, among others. In some embodiments, the sensor system 130 is incorporated with the operational systems 106. For example, one or more sensors of the sensor system 130 may be incorporated with the brake system 124 monitor characteristics of the host 302, such as deceleration.

Accordingly, the sensor system 130 is operable to sense a measurement of data associated with the device, the operating environment 100, the device environment, and/or the operational systems 106 and generate a data signal indicating said measurement of data. These data signals can be converted into other data formats (e.g., numerical) and/or used by the sensor module 102, the computing device 104, and/or the operational systems 106 to generate other data metrics and parameters. It is understood that the sensors can be any type of sensor, for example, acoustic, electric, environmental, optical, imaging, light, pressure, force, thermal, temperature, proximity, among others.

The sensor module 102, the computing device 104, and/or the operational systems 106 are also operatively connected for computer communication to the network 132. The network 132 is, for example, a data network, the Internet, a wide area network (WAN) or a local area (LAN) network. The network 132 serves as a communication medium to various remote devices (e.g., databases, web servers, remote servers, application servers, intermediary servers, client machines, other portable devices). Using the system and network configuration discussed above, anomalous events can be detected. Detailed embodiments describing exemplary methods using the system and network configuration discussed above will now be discussed in detail.

II. Methods for Crowd Navigation

Referring now to FIG. 2, a method 200 for crowd navigation will now be described according to an exemplary embodiment. FIG. 2 will also be described with reference to FIGS. 1 and 3A-5. For simplicity, the method 200 will be described as a sequence of elements, but it is understood that the elements of the method 200 can be organized into different architectures, blocks, stages, and/or processes.

At block 202 the method 200 includes the statistical module 120 identifying a number of agents. Turning to FIG. 3A, the agents 304-312 are entities moving in a physical environment of the host 302. Depending on the movement, path planning, and/or capability identify the position of the agents 304-312. The agents 304-312 may be biological entities (humans, animals, insects), vehicles, robots, etc. The agents 304-312 may be identified based on sensor data 110 including, visual data, motion data, and physiological data, among others. In this manner, the statistical module 120 may detect or identify one or more of the entities, objects, obstacles, hazards, and/or corresponding attributes or characteristics including agent identification, a position or a location associated with the agents 304-312, such as a lane location, coordinates, position, size of the agents 304-312, trajectory, velocity, acceleration, etc.

The statistical module 120 may also model the characteristics and attributes of the agents 304-312 relative to the host 302, as shown in FIG. 3B. The characteristics and attributes determined by the statistical module 120 of agents 304-312 relative to the host 302 may be classified as first order interactions. The first order interactions may include, for example, an agent identification, how the trajectories of the agents 304-312 coincide with the path planning of the host 302, speeds of the agents 304-312 relative to the host 302, distances of the agents 304-312 from the host 302, a bearing or direction of travel of the agents 304-312 relative to the host 302, acceleration of the agents 304-312 relative to the host 302, etc. Moreover, the statistical module 120 may determine if one or more of the agents 304-312 is attempting to communicate with the host. Interplay between the host 302 and one or more of the agents 304-312 may be determined based on eyelid movement, head movement and/or mouth movement of the agents 304-312. The statistical module 120 may measure the degree of head movement such as the tilt of the head of the agents 304-312 relative to the host 302 or the angle of the head of the agents 304-312 relative to the host 302.

In some embodiments, the host 302 may measure the relative gap between agents 308 and 310 relative to the host 302 as well as the relative gap between agents 310 and 312 relative to the host 302. The statistical module 120 may determine that the gap between the agents 308 and 310 is larger and thus better able to accommodate the host 302, as shown in FIG. 3B.

Furthermore, the statistical module 120 may determine attributes or characteristics of the agents 304-312 relative to one another, as shown in FIG. 3C. The characteristics and attributes determined by the statistical module 120 of agents 304-312 relative to one another may be classified as second order interactions. The first order interactions may include, for example, position or a location of the agents 304-312 relative to one another, speeds of the agents 304-312 relative to one another, distances of the agents 304-312 relative to one another, a bearing or direction of travel of the agents 304-312 relative to one another, acceleration of the agents 304-312 relative to one another, distances of the agents 304-312 relative to one another, a, such as a lane location, coordinates, etc. The statistical module 120 may determine if the agents 304-312 are communicating with one another. Interplay between one or more of the agents 304-312 may be determined based on eyelid movement, head movement and/or mouth movement of the agents 304-312. The statistical module 120 may measure the degree of head movement such as the tilt of the head of the agents 304-312 relative to other agents or the angle of the head of the agents 304-312 relative to other agents. For example, the statistical module 120 may identify the interplay of the agents 310 and 312 based on the gap between the agents 310 and 312, corresponding mouth movements between the agents 310 and 312.

In one or more embodiments, the statistical module 120 may identify white lines and hard shoulders of a roadway or road segment to facilitate lane recognition. In another embodiment, the statistical module 120 may identify infrastructure of the physical environment. Further, the statistical module 120 may identify or classify an agent of the agents 304-312 as different types of agents, for example, a pedestrian, vehicle, a cyclist, etc. The different types may be based on the speed at which the agent moves, the size of the agent, and/or other sensor data 110.

The sensor module 102 receives sensor data 110. The sensor data 110 may be received from the sensor system 130, remote devices (e.g., via the bus 108 and/or the communications interface 118), and/or a biological entity. The sensor data 110 may include a video sequence or a series of images, user inputs, and/or data from the operational systems 106, such as data from a Controller Area Network (CAN) bus including as pedal pressure, steer angle, etc. The sensor system 130 may include one or more radar units, image capture components, sensors, cameras, gyroscopes, accelerometers, scanners (e.g., 2-D scanners or 3-D scanners), or other measurement components. In some embodiments, the sensor data 110 is augmented as additional sensor data from other sources is received. For example, the data from the CAN bus may be augmented by information the agents 304-312, the types of agent, and image/video data, among others.

At block 204 the method 200 includes calculating a Gaussian Process for each agent of the agents 304-312 and the host 302. A Gaussian Process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. Here the Gaussian Processes may be defined as X to be the state of the agents 304-312 and the host 302. For example, X could be

² for planar navigation. The measurements z_(1:t) ^(R) of the trajectory of the host 302 f^(R): t ∈

→X and n_(t) measurements z_(1:t) ¹, . . . , z_(1:t) ^(n) ^(t) of the human trajectories f=[f¹, f^(n) ^(t) ]: t ∈

→X, where f represents the crowd. The functions f^(R) and f may be governed by p(f^(R)|z_(1:t) ^(R)) and p(f^(i)|z_(1:t) ^(f) ^(i) ) for each i. Herein, the shorthand z_(1:t) ^(f)=[z¹:t^(f) ¹ , . . . , z_(1:t) ^(f) ^(n) ^(t)] may be used. Further, fR, f^(i) ∈

(X), the function space over X and the shorthand

(x|μ, Σ)≡GP(x|μ, Σ) may be used by the statistical module 120 to define the Gaussian processes of the agents 304-312 and the host 302 as:

$\begin{matrix} {{{p\left( f^{i} \middle| z_{1:t}^{f^{i}} \right)} = {\Sigma_{k_{i} = 1}^{N_{t}^{f^{i}}}w_{k_{i}}^{f^{i}}\left( {\left. f^{i} \middle| \mu_{k_{i}}^{f^{i}} \right.,\Sigma_{k_{i}}^{f^{i}}} \right)}}{{{for}\mspace{14mu} {each}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} {agents}\mspace{14mu} 304\text{-}312},{and}}{{p\left( f^{R} \middle| z_{1:t}^{R} \right)} = {\sum_{l = 1}^{N_{t}^{R}}{w_{}^{R}\left( {\left. f^{R} \middle| \mu_{}^{R} \right.,\Sigma_{}^{R}} \right)\mspace{14mu} {for}\mspace{14mu} {the}\mspace{14mu} {host}\mspace{14mu} 302.}}}} & \left( {{Eq}.\mspace{11mu} 01} \right) \end{matrix}$

Furthermore, as the statistical module 120 may continually calculate the Gaussian processes as sensor data 110 is received. For clarity, we express time:

≡

(t), . . . ,

w_(k_(n_(t)))^(f^(n_(t))) ≡ w_(k_(n_(t)))^(f^(n_(t)))(t)

and μ≡μ(t), Σ≡Σ(t). In this manner, the Gaussian processes may be defined as follows:

Definition 1: Let N_(f) _(R) _(,l)=N(f^(R)|

,

). For each f^(i), let N_(f,v)p=N(f^(p)|μ_(v) ^(p), Σ_(v) ^(p)) where v=k_(i) and p ∈ {1, . . . , n_(t)}. For example p(f^(R)|z_(1:t) ^(R))=

.

Definition 2: The intents of the agents 304-312 and the host 302 are given by μ_(k) _(i) ^(f) ^(i) ,

respectively. If N_(t) ^(f) ^(i) or N_(t) ^(R) is greater than 1, intent ambiguity is present. Intent preferences are w_(k) ^(f) ^(i) and

.

Definition 3: Flexibility is the willingness of the agents 304-312 to compromise their intent. Mathematically, the flexibility of intent μ is Σ.

In Equation 0.1, intent preferences are the data likelihood, e.g. w_(k) _(R) ^(R)=

(f^(R)=z_(1:t) ^(R)μ_(k) _(R) ^(R), Σ_(k) _(R) ^(R)). To generate the Gaussian processes the data z_(1:t) ^(R), z_(1:t) ^(f) is used. Let p(f^(i)|z_(1:t) ^(f) ^(i) )=Σ_(k) _(i) _(=L,R)w_(k) _(i) ^(f) ^(i)

_(f) _(i) _(,k) _(i) . Because N_(t) ^(f) ^(i) =2, intent ambiguity is present over left and right intents. Flexibility is motivated by the following. Draw an agent sample x and evaluate according to

(f^(i)=x|μ, Σ). For large covariance, large deviation from intent returns nontrivial probabilities (e.g., large flexibility); for small covariance, large deviation from intent returns vanishing probability.

In this manner, the crowd navigation uses a joint host-crowd density p(f^(R), f|z_(1:t) ^(R), z_(1:t) ^(f)) to generate the action u_(t)=f_(t+1) ^(R)* at time t according to:

[f ^(R) , . . . , f ^(n) ^(t) ]*=arg max p(f ^(R) , ff ^(R) , . . . , f ^(n) ^(t) |z _(1:t) ^(R) , z _(1:t) ^(f))   (Eq. 0.2)

At block 206 of the method 200, an objective function is determined based on the intent and flexibility. For example, suppose the intent of the host 302 is to avoid a collision with the agent 312 while also continuing to move in as direct a path as possible. In this manner, the intent can be a balance between competing goals. The flexibility is the degree to which the host 302 can deviate from the intent. In this manner, the Gaussian processes define the intent and flexibility for the host 302 and the agents 304-312 as the mean and variance, respectively, of the function ψ.

Starting with the Gaussian processes, a set of principles is derived that the objective function—the function coupling the agent models—conforms to. Accordingly, the objective function is determined based on desired safety and efficiency properties and an optimization routine to find μ_(t)=f_(t+1) ^(R)*.

$\begin{matrix} {{p\left( {f^{R},\left. f \middle| z_{1:t}^{R} \right.,z_{1:t}^{f}} \right)} = {{{\psi \left( {f^{R},f,\gamma} \right)}{p\left( f \middle| z_{1:t}^{f} \right)}{p\left( f^{R} \middle| z_{1:t}^{R} \right)}} = {\prod\limits_{i = 1}^{n_{t}}{{\psi \left( {f^{R},f^{i},\gamma} \right)}{p\left( f^{i} \middle| z_{1:t}^{f^{i}} \right)}{p\left( f^{R} \middle| z_{1:t}^{R} \right)}}}}} & \left( {{Eq}.\mspace{11mu} 0.3} \right) \end{matrix}$

The function ψ(f^(R), f, γ), γ ∈

is a product of pairwise objective functions ψ(f^(R), f^(i), γ) modulated by p(f^(i)|z_(1:t) ^(f) ^(i) ). We generalize the bi-agent interaction case, for example, for a host 302 and two interacting agents, such as the agents 306 and 308, such as:

p(f ^(R) , f|z _(1:t))=ψ(f ^(R) , f ¹, γ)ψ(f ^(R) , f ², γ)×p(f ^(R) |z _(1:t) ^(R))p(f ¹ |z _(1:t) ^(f) ¹ )p(f ² |z _(1:t) ^(f) ² ).

Now suppose that the hosts is interacting with three agents, such as the agents 306, 308, and 310, then the objective function is given by:

ψ(f^(R), f¹, γ)ψ(f^(R), f², γ)ψ(f^(R), f³, γ)×ψ(f¹, f², γ)ψ(f¹, f³, γ)ψ(f², f³, γ)

Generalized the objective function is given by:

${p\left( {f^{R},\left. f \middle| z_{1:t}^{R} \right.,z_{1:t}^{f}} \right)} = {{{\psi \left( {f^{R},f,\gamma} \right)}{p\left( f^{R} \middle| z_{1:t}^{R} \right)}{p\left( f \middle| z_{1:t}^{f} \right)}} = {\prod\limits_{i = 1}^{n_{t}}{{\psi \left( {f^{R},f^{i},\gamma} \right)}{\prod\limits_{j > i}^{n_{t}}{{\psi \left( {f^{i},f^{j},\gamma} \right)}{p\left( f^{R} \middle| z_{1:t}^{R} \right)}{p\left( f^{i} \middle| z_{1:t}^{f^{i}} \right)}}}}}}$

To determine statistically valid forms of ψ(f^(i), f^(j), γ), where i ∈ {R, 1, . . . , n_(t)}. Since p(f^(i)|z_(1:t) ^(i)), p(f^(j)|z_(1:t) ^(j)) encode intention and flexibility information, the means and covariances capture inter-agent intention and flexibility that is specific to and influenced by agent interaction. If the interaction function has finite support (e.g.,

${\psi \left( {f^{i},f^{j},\gamma} \right)} = {\Pi_{t = 1}^{T}\left\lbrack {1 - {\exp \left( {{- \frac{1}{2\gamma}}\left( {f_{t}^{i} - f_{t}^{j}} \right)} \right)}} \right\rbrack}$

where γ>0), joint flexibility is altered in a static and generic way; any interaction probability mass encoded in ψ(f^(i), f^(j), γ) alters the agent-specific flexibilities (e.g., agent 1 and agent 2 are flexible with each other in a specific way, which is already captured in p(f^(i) z_(1:t) ^(f) ^(i) ), p(f^(j) z_(1:t) ^(f) ^(j) ). To preserve the statistics of p(f^(i)|z_(1:t) ^(f) ^(i) ), p(f^(j)|z_(1,t) ^(f) ^(j) ) we introduce the following transform, where f_(d) ^(i)˜p(f^(i)|z_(1:t) ^(f) ^(t) ), f_(7n) ^(j)˜p(f^(j)|z_(1:t) ^(f) ^(j) ):δ(f^(R), f^(i))

Suppose

${{\overset{¯}{\delta}\left( {f_{d}^{i},f_{m}^{j}} \right)} \equiv {\lim\limits_{\gamma\rightarrow 0}\left\lbrack {\Pi_{\tau = 1}^{T}\left( {1 - {\exp \left\lbrack {{- \frac{1}{2\gamma}}\left( {{f_{d}^{i}(\tau)} - {f_{m}^{j}(\tau)}} \right)^{2}} \right\rbrack}} \right)} \right\rbrack}} = \left\{ {\begin{matrix} {{1\ {if}\ {\nexists{t \in {\left\lbrack {1,\ T} \right\rbrack \ {such}\mspace{14mu} {that}\ {f_{d}^{i}(t)}}}}} = {f_{m}^{j}(t)}} \\ {{0\ {if}\ {\exists{t \in {\left\lbrack {1,\ T} \right\rbrack \ {such}{\; \mspace{9mu}}{that}\ {f_{d}^{i}(t)}}}}} = {f_{m}^{j}(t)}} \end{matrix},} \right.$

Given ψ(f^(i), x, y) has finite support over f^(i) then it may alter agent flexibility via Σ_(k) _(i) ^(f) ^(i) of

_(f) _(i) _(,k) _(i) . Let ψ(f^(i), f^(j), γ)=c_(k) _(i) _(,k) _(j) ; then using Equation 0.1 and 0.3, is given by

p(f^(i), f^(j)|z_(1 : t)) = Σ_(k = 1)^(N_(t)^(f^(i)))Σ_(k = 1)^(N_(t)^(f^(j)))c_(k_(i), k_(j))w_(k_(λ))^(f^(i))w_(k_(ϰ))^(f^(j))_(f^(i), k_(i))N_(f jk j).

Although the joint undergoes distortion from the effect of c_(k) _(i) _(,k) _(j) on the component weights w_(k) _(i) ^(f) ^(i) w_(k) _(j) ^(f) ^(j) , no individual Σ_(k) _(i) ^(f) ^(i) or Σ_(k) _(j) ^(f) ^(j) in component

_(f) _(R) _(,l)

_(f) _(i) _(,k) _(i) is altered. Thus, the flexibility of p(f^(i)|z_(1:t) ^(f) ^(i) ), p(f^(j)|z_(1:t) ^(f) ^(j) ) is preserved under c_(k) _(i) _(,k) _(j) . Finally, let ψ(f^(R), f^(i), γ)=δ(f^(R), f^(i)). Let f_(a) ^(i)˜p(f^(i)|z_(1:t) ^(f) ^(i) ), f_(b) ^(R)˜p(f^(R)|z_(1:t) ^(R)), η_(a) ^(f) ^(i) =p(f^(i)=f_(a) ^(i)|z_(1:t) ^(f) ^(i) ), and η_(b) ^(R)=p(f^(R)=f_(b) ^(R)|z_(1:t) ^(R)). Then, for all b, p(f_(b) ^(i), f_(a) ^(j)|z_(1:t))=δ(f_(b) ^(i), f_(a) ^(j))η_(b) ^(f) ^(i) η_(a) ^(f) ^(j) . If

t ∈ [1, T] such that f_(b) ^(i)(t)=f_(a) ^(j)(t), then δ(f_(b) ^(i), f_(a) ^(j))η_(b) ^(f) ^(i) η_(a) ^(f) ^(j) =η_(b) ^(f) ^(i) η_(a) ^(f) ^(j) ; otherwise, it is zero. Thus, δ(f^(i), f^(j)) respects the agent flexibility data contained in p(f^(i)|z_(1:t) ^(f) ^(i) ). The same argument can be made for p(f^(i)|z_(1:t) ^(f) ^(i) ); thus, δ(f^(i), f^(j)) respects the flexibility data in p(f^(i)|z_(1:t) ^(f) ^(i) ) and p(f^(j)|z_(1:t) ^(f) ^(j) ). Using a trajectory basis so that p(f^(i), f^(j) z_(1:t))=Σ_(g) ^(G)w_(g)δ([f^(i), f^(j)]−[f^(i), f^(j)]_(g)) then δ(f^(i), f^(j)) would be applicable: discard intersecting samples. Since the interaction of two GPs is probabilistic, a “coupling” probability is appropriate.

At block 208 the method 200 includes the model module 122 generating a model for the agents the 304-312 relative to the host. The model is generated by applying the objective function to the set of Gaussian processes. For Gaussian Processes, the probability of host-agent collision does not only involve time-aligned terms: z_(k) _(i) _(,k) _(j) _(,t) ⁻¹=∫

(z|μ_(k) _(i) _(,t) ^(f) ^(t) , σ_(f) _(i) _(,k) _(i) ^(t,t))

(x|μ_(k) _(j) _(,t) ^(f) ^(j) , σ_(f) _(j) _(,k) _(j) ^(t,t)) where μ_(k) _(i) _(,t) ^(f) ^(i) =μ_(k) _(i) ^(f) ^(i) (t), μ_(k) _(j) _(,t) ^(f) ^(j) =μ_(k) _(j) ^(f) ^(j) (t) ∈

² and i ∈ {R, 1, . . . , n_(t)}; σ^(t,t) t is the t^(th) diagonal of Σ. Since Σ_(l) ^(R), Σ_(k) _(i) ^(f) ^(i) are dense, positions are correlated via covariance off diagonals and ∫

(x|μ_(k) _(i) _(,t) ^(f) ^(i) , σ_(f) _(i) _(,k) _(i) ^(t,τ))

(x|μ_(f) _(i) _(,k) _(i) ^(t,τ))=w_(k) _(i) _(,k) _(j) _(,t)×exp[−½(μ_(k) _(i) _(,t) ^(f) ^(i) −μ_(k,τ) ^(f) ^(j) )^(T)(σ_(k) _(i) _(+k) _(j) ^(t,τ)I)⁻¹(μ_(k) _(i) _(,t) ^(f) ^(λ) )]≡w_(k) _(i) _(,k) _(j) _(,t)Z_(k) _(i) _(,k) _(j) _(,t,τ) ⁻¹, contributes to collision probability (σ_(k) _(i) _(+k) _(j) ^(t,τ) is the (t, τ)'th element of Σ_(k) _(i) _(+k) _(j) =Σ_(k) _(λ) ^(f) ^(i) +Σ_(k) _(j) ^(f) ^(j) and w_(k) _(i) _(,k) _(j) _(,t)=(2πσ_(k) _(λ) _(+k) _(j) ^(t,τ))^(−1/2). Since z_(k) _(i) _(,k) _(j) _(,t,τ) ⁻¹ is the coupling between f_(t) ^(i) and f_(τ) ^(j), the value Π_(τ=1) ^(T)(1−z_(k) _(i) _(,k) _(j) _(,t,τ) ⁻¹) is the decoupling between agent i at t—f_(t) ^(i)—and the trajectory of agent j, f^(j).

Definition 5. The symbol

(¬κ)—the probability of not colliding—represents the decoupling of

and

by

${{\mathbb{P}}\left( {\kappa} \right)} = {\prod\limits_{r = 1}^{T}{\prod\limits_{\tau = 1}^{T}\left( {1 - Z_{k_{i},k_{j},t,\tau}^{- 1}} \right)}}$

Definition 6. The transform P¬κ measures how decoupled the host and agent GPs

_(f) _(η) _(,k) _(i) and

_(fjk) _(j) are:

P _(¬κ):

_(f) _(i) _(,k) _(i) →Π_(t=1) ^(T)Π_(τ=1) ^(T)(1−z _(k) _(i) _(,k) _(j) _(,t,τ) ⁻¹)

_(f) _(i) _(,k) _(i)

_(fjk)=∧^(k) ^(?) ^(,k) ^(j)

_(f) _(i) _(,k) _(t)

_(fjk) _(j) .

where ∧^(k) ^(i) ^(,k) ^(j) ≡Π_(t=1) ^(T)Π_(τ=1) ^(T)(1−z _(k) _(i) _(,k) _(j) _(,t,τ) ⁻¹).

Agent models may be defined as f_(t) ^(i)=h(f_(t−1) ^(i), η_(t)), η_(t)˜

(0, σ_(η) _(t) ). This induces a diagonal trajectory covariance matrix: agent flexibility at t is decoupled from flexibility at t+1. Accordingly, the objective function is applied to the set of Gaussian processes for the host 302 and the agents 304-312. The objective function may be applied to a Gaussian process by:

$\begin{matrix} {{p\left( {f^{R},\left. f^{i} \middle| z_{1:t} \right.} \right)} = {{P_{\kappa}\left\lbrack {\sum\limits_{ = 1}^{N_{t}^{R}}{w_{}^{R}_{f^{R},}{\sum\limits_{k_{i} = 1}^{N_{t}^{f^{i}}}{w_{k_{i}}^{f^{i}}_{f^{i},k_{i}}}}}} \right\rbrack} = {\sum\limits_{ = 1}^{N_{t}^{R}}{\sum\limits_{k_{i} = 1}^{N_{t}^{f^{i}}}\; {\Lambda_{l,}k_{i}w_{}^{R}w_{k_{i}}^{f^{i}}_{f^{R},}_{f^{\overset{.}{x}},k_{i}}}}}}} & \left( {{Eq}.\mspace{11mu} 0.7} \right) \end{matrix}$

Eq. 0.7 can be generalize IGP to n_(t)≥1, such that for a multi-agent second order Gaussian process can be given by

${p\left( {f^{R},\left. f \middle| z_{1:t} \right.} \right)} = {{p_{\kappa}^{IGP}\left\lbrack {\sum_{k_{R} = 1}^{N_{t}^{R}}{W_{k_{R}}^{R}_{f^{R},k_{R}}{\prod_{i = 1}^{n_{t}}{\sum_{k_{i} = 1}^{N_{t}^{f^{i}}}{w_{k_{i}}^{f^{i}}_{f^{i},k_{i}}}}}}} \right\rbrack} = {p_{\kappa}^{IGP}\left\lbrack {{\sum_{k_{R} = 1}^{N_{t}^{R}}{w_{k_{R}}^{R}_{f^{R},k_{B}} \times {\sum_{k_{1} = 1}^{N_{t}^{f^{1}}}{w_{k_{1}}^{f^{1}}N_{f^{1},k_{1}} \times \ldots \times {\sum_{k_{n_{t}} = 1}^{N_{t}^{f^{n}t}}{w_{k_{n_{t}}}^{f^{{r\; \iota},}}_{f^{},,k_{n_{t}}}}}}}}} = {\sum_{\eta = 1}^{N_{BTG}}{\left\lbrack {w\Lambda} \right\rbrack_{\eta}\left\lbrack {_{f^{R}}_{f^{1}}\mspace{14mu} \ldots \mspace{14mu} _{f^{n_{t}}}} \right\rbrack}_{\eta}}} \right.}}$

where N_(BIG)=N_(t) ^(R)Π_(i=1) ^(n) ^(t) N_(t) ^(f) ^(i) and η enumerates all products of robot and agent GPs. If 1>R, then the coefficients pa [wΛ]_(η)=Π_(i=R) ^(n)w_(η) ^(f) ^(t) Π_(j>i) ^(n)Λ_(η) ^(k) ^(i) ^(,k) ^(j) weight each GP basis element [

_(f) _(R)

_(f) ₁ . . .

_(f) _(n) _(t)]_(η)=

_(f) _(R) _(,η)

_(f) ₁ _(,η) . . .

_(f) _(n) _(,η) according to p_(¬κ) ^(IGP). The operator p_(¬κ) ^(IGP) operates pairwise p_(¬κ) ^(IGP) ^(≡) (p_(¬κ))^(Σ) ^(m=0) ^(n) ^(t) ⁻¹ ^((n) ^(t) ^(−m))=p_(¬κ) _(oP) _(¬Λ) _(o . . . oP) _(¬κ), on host-agent pairs and as well as agent-pairs. Accordingly, the objective function for modeling first order interactions and second order interactions may be given by.

$\left( {f^{R},\left. f \middle| z_{1:t} \right.} \right) = {{p_{\kappa}^{IGP}\left\lbrack {\sum_{k_{R} = 1}^{N_{t}^{R}}{W_{k_{R}}^{R}_{f^{R},k_{R}}{\prod_{i = 1}^{n_{t}}{\sum_{k_{i} = 1}^{N_{t}^{f^{i}}}{w_{k_{i}}^{f^{i}}_{f^{i},k_{i}}}}}}} \right\rbrack} = {p_{\kappa}^{IGP}\left\lbrack {{\sum_{k_{R} = 1}^{N_{t}^{R}}{w_{k_{R}}^{R}_{f^{R},k_{B}} \times {\sum\limits_{k_{1} = 1}^{N_{t}^{f^{1}}}{w_{k_{1}}^{f^{1}}N_{f^{1},k_{1}} \times \ldots \times {\sum\limits_{k_{n_{t}} = 1}^{N_{t}^{f^{n}t}}\; {w_{k_{n_{t}}}^{f^{{r\; \iota},}}_{f^{},,k_{n_{t}}}}}}}}} = {\sum_{\eta = 1}^{N_{BTG}}{\left\lbrack {w\Lambda} \right\rbrack_{\eta}\left\lbrack {_{f^{R}}_{f^{1}}\mspace{14mu} \ldots \mspace{14mu} _{f^{n_{t}}}} \right\rbrack}_{\eta}}} \right.}}$

where N_(BIG)=N_(t) ^(R)Π_(i=1) ^(n) ^(t) N_(t) ^(f) ^(i) and η enumerates all products of host and agent Gaussian processes. The coefficients [Λw]_(η)=Λ_(η) ^(R,f) ¹ . . . Λ_(η) ^(R,f) ^(n) ^(t)w_(η) ^(R)w_(η) ^(f) ¹ . . . w_(η) ^(f) ^(n) ^(t) weight each Gaussian process basis element [

_(f) _(R)

_(f) ₁ . . .

_(f) _(n) _(t)]_(η)=

_(f) _(R) _(,η)

_(f) ₁ _(,η) . . .

_(f) _(n) _(t,η) according to p_(¬κ) ^(IGP). The operator P_(¬κ) ^(IGP) operates pairwise P_(¬κ) ^(IGP)≡(P_(¬κ) ^(IGP))^(n) ^(t) =P_(¬κ)∘P_(¬κ)∘ . . . ∘P_(¬κ) such that the agents 304-312 can be measured with respect to the host 302.

Instead of brute force enumeration, the N*«N_(BIG) modes are determined that capture Equation 0.8, in a process called optimal shaping such that a convex configuration of the number of agent in the physical environment is determined. Accordingly, the modes with the most likely probability are determined.

Firstly, μ_(l) ^(R), μ_(k) _(i) ^(f) ^(i) are treated as functions

, x_(f) _(i) _(,k) _(i) ∈

(

)→

² mapping time to (x, y) position and search for the

, x_(f) _(i) _(,k) _(i) * that optimize [Λw]_(η) such that:

$w_{x_{R,}} = {\left( {\left. x_{R,} \middle| \mu_{}^{R} \right.,{{\Sigma_{)}^{R}w_{x_{f^{i},k_{i}}}} = {{\left( {\left. x_{f^{i},k_{i}} \middle| \mu_{k_{i}}^{f^{i}} \right.,{\Sigma_{k_{i})}^{f^{i}}Z_{x_{R,,t,}x_{f^{i},k_{i},\tau}}^{- 1}}} \right)} = {{{\exp \left\lbrack {{- \frac{1}{2}}\left( {{x_{R,,t} - x_{f^{i}}},{ki},_{T}} \right){T\left( {\sigma_{ + k_{i}}^{t,\tau}{}} \right)}^{- 1}\left( {x_{R,,t} - x_{{f^{i}k_{i}},\tau}} \right)} \right\rbrack}{where}\text{}x_{R,,t}} = {x_{R,}(t)}}}},{x_{f^{i},k_{i},\tau} = {{x_{f^{i},k_{i}}(\tau)} \in {{\mathbb{R}}^{2}.}}}} \right.}$

If

x_(f) = [x_(f¹, k₁,)x_(f^(n)t, k_(n_(t)))],

then an objective function of the interacting function may be given by

λ_(n_(t))(x_(R, l), x_(f)) ≡ w_(x_(R, ))Π_(i = 1)^(n_(t))Π_(t = 1)^(T)Π_(τ = 1)^(T)(1 − Z_(x_(R, , t,)x_(f^(i), k_(i), τ))⁻¹)w_(x_(f^(i), k_(i)))

A logarithm can be used to reduce the optimization computational burden and increase numerical accuracy such that:

$\begin{matrix} {{\log {\lambda_{n_{t}}\left( {x_{R,l},x_{f}} \right)}} = {{\sum\limits_{i = 1}^{n_{t}}{\sum\limits_{t = 1}^{T}{\sum\limits_{\tau = 1}^{T}{\log \left( {1 - Z_{x_{R,,t,}x_{f^{i},k_{i},\tau}}^{- 1}} \right)}}}} - {\frac{1}{2}\left( {x_{R,l} - \mu_{}^{R}} \right)^{T}\left( \Sigma^{R} \right)^{- 1}\left( {x_{R,l} - \mu_{}^{R}} \right)} - {\Sigma_{i = 1}^{n_{t}}\frac{1}{2}\left( {x_{f^{\overset{.}{t}},k_{i}} - \mu_{k_{i}}^{f^{i}}} \right)^{T}\left( \Sigma^{f^{i}} \right)^{- 1}\left( {x_{f^{i},k_{2}}.{- \mu_{k_{i}}^{f^{i}}}} \right)}}} & \left( {{Eq}.\mspace{11mu} 0.9} \right) \end{matrix}$

So that the convex configuration of the agents 304-312 in the physical environment is given by:

[

, x _(f)*]=arg, max log [λ_(n) _(t) (x _(R,l) , x _(f))]  (Eq. 0.10)

The objective function defines the intent. Returning to the example given above, suppose the intent of the host 302 is to avoid a collision with the agent 312 while also continuing to move in as direct a path as possible. Avoiding a collision and encouraging cooperation may be given by:

${\alpha_{n_{t}}\left( {x_{R,},x_{f}} \right)} = {\sum\limits_{i = 1}^{n_{t}}{\prod\limits_{t = 1}^{T}{\prod\limits_{\tau = 1}^{T}{\log \left( {1 - Z_{x_{R,,t,}x_{f^{i},k_{i},\tau}}^{- 1}} \right)}}}}$

while continuing to move in as direct a path, such as a straight line may be given by:

β_(n) _(t) (x _(R,l) , x _(f))=−½(x _(R,l)−

)^(T)(

)⁻¹(

−

)=−½Σ_(i=1) ^(n) ^(t) (x _(f) _(i) _(,k) _(i) n _(t)−μ_(k) _(i) ^(f) ^(i) )^(T)(Σ_(k) _(i) ^(f) ^(i) )⁻¹(x _(f) _(i) _(,k) _(i) −μ_(k) _(i) ^(f) ^(i) ).

Accordingly, while α_(n) _(t) (x_(R,l), x_(f)) encourages a cooperative protocol and prevents collisions, β_(n) _(t) (x_(R,l), x_(f)) penalizes actions that deviate from agent intent. Unless the optimal values for β_(n) _(t) (x_(R,l), x_(f)) already optimize λ_(n) _(t) (x_(R,l), x_(f)), α_(n) _(t) (x_(R,l), x_(f)) moves x_(R,l), x_(f) away from intent.

To generate the convex configuration, the optimizations are seeded with (μ^(R), μ^(f))±(0, [σ^(R), σ^(f)], 2[σ^(R), σ^(f)], 3[σ^(R), σ^(f)]), where σ^(R)=√{square root over (diag(Σ^(R)))}, σ^(f)=√{square root over (diag(Σ^(f)))}. The optimizations provide insight about the agents 304-312. In particular, the optimizations may be calculated for each agent of the agents 304-312 and the host 302 and then computed the effective sample size of n_(t) host-agent pairs to determine how many agents are statistically significant to the optimization. In this manner, a subset of the agents 304-312 can be determined for each time step.

Once the convex configuration is determined it may be used by the brake system 124, the path planning module 126, the notification system 128, and the sensor system 130 to alter the functioning of the host 302, the agents 304-312, and/or infrastructure, such as the sensor module 102. For example, the brake system 124 may transmit an instruction for the host 302 and/or the agents 304-312 to brake based on the convex configuration. The path planning module 126 may plan a path or adjust a path of the host 302 based on the convex configuration modeled by crowd.

In another embodiment, the notification system 128 may notify the agents 304-312 based on the convex configuration, for example, the agents 304-312 may notified of the path of the host 302. In another embodiment, the sensor system 130 may adjust the manner and/or location that is sensed within the physical environment.

In this manner, Gaussian Processes can be used to model the joint interaction between the host 302 and the agents 304-312 as well as the agents 304-312 among each other. The Gaussian Processes could model an infinite number of trajectories, however most trajectories are not likely. For example, it is not likely that an agent will zig-zag to a target. Instead, the trajectories are grouped into lanes and optimize over the function space and a Gaussian Processes is calculated for each agent based on the host-agent pairs as well as the agent-agent pairs. The Gaussian processes are multiplied by an objective function having a mean indicative of intent and a variance indicative of flexibility such that the Gauss. Thus, the means become the basis set for a distribution of the Gaussian Processes. In this manner, the intent can be used to identify high value lanes. Further we allow the constants to vary based on the flexibility.

Still another aspect involves a computer-readable medium including processor-executable instructions configured to implement one aspect of the techniques presented herein. An aspect of a computer-readable medium or a computer-readable device devised in these ways is illustrated in FIG. 4, wherein an implementation 400 includes a computer-readable medium 408, such as a CD-R, DVD-R, flash drive, a platter of a hard disk drive, etc., on which is encoded computer-readable data 406. This encoded computer-readable data 406, such as binary data including a plurality of zero's and one's as shown in 406, in turn includes a set of processor-executable computer instructions 404 configured to operate according to one or more of the principles set forth herein. In this implementation 400, the processor-executable computer instructions 404 may be configured to perform a method 402, such as the method 200 of FIG. 2. In another aspect, the processor-executable computer instructions 404 may be configured to implement a system, such as the operating environment 100 of FIG. 1. Many such computer-readable media may be devised by those of ordinary skill in the art that are configured to operate in accordance with the techniques presented herein.

As used in this application, the terms “component”, “module,” “system”, “interface”, and the like are generally intended to refer to a computer-related entity, either hardware, a combination of hardware and software, software, or software in execution. For example, a component may be, but is not limited to being, a process running on a processor, a processing unit, an object, an executable, a thread of execution, a program, or a computer. By way of illustration, both an application running on a controller and the controller may be a component. One or more components residing within a process or thread of execution and a component may be localized on one computer or distributed between two or more computers.

Further, the claimed subject matter is implemented as a method, apparatus, or article of manufacture using standard programming or engineering techniques to produce software, firmware, hardware, or any combination thereof to control a computer to implement the disclosed subject matter. The term “article of manufacture” as used herein is intended to encompass a computer program accessible from any computer-readable device, carrier, or media. Of course, many modifications may be made to this configuration without departing from the scope or spirit of the claimed subject matter.

FIG. 5 and the following discussion provide a description of a suitable computing environment to implement aspects of one or more of the provisions set forth herein. The operating environment of FIG. 5 is merely one example of a suitable operating environment and is not intended to suggest any limitation as to the scope of use or functionality of the operating environment. Example computing devices include, but are not limited to, personal computers, server computers, hand-held or laptop devices, mobile devices, such as mobile phones, Personal Digital Assistants (PDAs), media players, and the like, multiprocessor systems, consumer electronics, mini computers, mainframe computers, distributed computing environments that include any of the above systems or devices, etc.

Generally, aspects are described in the general context of “computer readable instructions” being executed by one or more computing devices. Computer readable instructions may be distributed via computer readable media as will be discussed below. Computer readable instructions may be implemented as program modules, such as functions, objects, Application Programming Interfaces (APIs), data structures, and the like, that perform one or more tasks or implement one or more abstract data types. Typically, the functionality of the computer readable instructions are combined or distributed as desired in various environments.

FIG. 5 illustrates a system 500 including an apparatus 512 configured to implement one aspect provided herein. In one configuration, the apparatus 512 includes at least one processing unit 516 and memory 518. Depending on the exact configuration and type of computing device, memory 518 may be volatile, such as RAM, non-volatile, such as ROM, flash memory, etc., or a combination of the two. This configuration is illustrated in FIG. 5 by dashed line 514.

In other aspects, the apparatus 512 includes additional features or functionality. For example, the apparatus 512 may include additional storage such as removable storage or non-removable storage, including, but not limited to, magnetic storage, optical storage, etc. Such additional storage is illustrated in FIG. 5 by storage 520. In one aspect, computer readable instructions to implement one aspect provided herein are in storage 520. Storage 520 may store other computer readable instructions to implement an operating system, an application program, etc. Computer readable instructions may be loaded in memory 518 for execution by processing unit 516, for example.

The term “computer readable media” as used herein includes computer storage media. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions or other data. Memory 518 and storage 520 are examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, Digital Versatile Disks (DVDs) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which may be used to store the desired information and which may be accessed by the apparatus 512. Any such computer storage media is part of the apparatus 512.

The term “computer readable media” includes communication media. Communication media typically embodies computer readable instructions or other data in a “modulated data signal” such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” includes a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal.

The apparatus 512 includes input device(s) 524 such as keyboard, mouse, pen, voice input device, touch input device, infrared cameras, video input devices, or any other input device. Output device(s) 522 such as one or more displays, speakers, printers, or any other output device may be included with the apparatus 512. Input device(s) 524 and output device(s) 522 may be connected to the apparatus 512 via a wired connection, wireless connection, or any combination thereof. In one aspect, an input device or an output device from another computing device may be used as input device(s) 524 or output device(s) 522 for the apparatus 512. The apparatus 512 may include communication connection(s) 526 to facilitate communications with one or more other devices 530, such as through network 528, for example.

Although the subject matter has been described in language specific to structural features or methodological acts, it is to be understood that the subject matter of the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example aspects. Various operations of aspects are provided herein. The order in which one or more or all of the operations are described should not be construed as to imply that these operations are necessarily order dependent. Alternative ordering will be appreciated based on this description. Further, not all operations may necessarily be present in each aspect provided herein.

As used in this application, “or” is intended to mean an inclusive “or” rather than an exclusive “or”. Further, an inclusive “or” may include any combination thereof (e.g., A, B, or any combination thereof). In addition, “a” and “an” as used in this application are generally construed to mean “one or more” unless specified otherwise or clear from context to be directed to a singular form. Additionally, at least one of A and B and/or the like generally means A or B or both A and B. Further, to the extent that “includes”, “having”, “has”, “with”, or variants thereof are used in either the detailed description or the claims, such terms are intended to be inclusive in a manner similar to the term “comprising”.

Further, unless specified otherwise, “first”, “second”, or the like are not intended to imply a temporal aspect, a spatial aspect, an ordering, etc. Rather, such terms are merely used as identifiers, names, etc. for features, elements, items, etc. For example, a first channel and a second channel generally correspond to channel A and channel B or two different or two identical channels or the same channel. Additionally, “comprising”, “comprises”, “including”, “includes”, or the like generally means comprising or including, but not limited to.

It will be appreciated that several of the above-disclosed and other features and functions, or alternatives or varieties thereof, may be desirably combined into many other different systems or applications. Also that various presently unforeseen or unanticipated alternatives, modifications, variations or improvements therein may be subsequently made by those skilled in the art which are also intended to be encompassed by the following claims. 

1. A system for crowd navigation of a host vehicle among a plurality of agents, comprising: a processor configured to receive sensor data; a statistical module configured to: identify a number of agents in a physical environment based on the sensor data; calculate a set of Gaussian processes, wherein the set of Gaussian processes includes a Gaussian Process for each agent of the number of agents; and determine an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents; and a model module configured to: generate a model of the number of agents by applying the objective function to the set of Gaussian processes, wherein the model includes a convex configuration of the physical environment.
 2. The system of claim 1, wherein the sensor data is received from a sensor system of the host.
 3. The system of claim 1, wherein the intent is based on a goal of the host, and wherein the flexibility is based on a willingness of the host to deviate from the goal.
 4. The system of claim 1, wherein the intent is a mean of the objective function and the flexibility is a covariance of the objective function.
 5. The system of claim 1, wherein the model module is further configured to perform optimal shaping to determine a subset of agents of the number of agents that are statistically significant, and wherein the convex configuration is based on the subset of agents.
 6. The system of claim 1, wherein the host is a robot, and wherein agents of the number of agents are humans.
 7. A method for crowd navigation of a host vehicle among a plurality of agents, comprising: identifying a number of agents in a physical environment based on sensor data from a host; calculating a set of Gaussian processes, wherein the set of Gaussian processes includes a Gaussian Process for each agent of the number of agents; determining an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents; and generating a model of the number of agents by applying the objective function to the set of Gaussian processes, wherein the model includes a convex configuration of the physical environment.
 8. The method of claim 7, wherein the sensor data is received from a sensor system of the host.
 9. The method of claim 7, wherein the intent is based on a goal of the host, and wherein the flexibility is based on a willingness of the host to deviate from the goal.
 10. The method of claim 7, wherein the intent is a mean of the objective function and the flexibility is a covariance of the objective function.
 11. The method of claim 7, further comprising: performing optimal shaping to determine a subset of agents of the number of agents that are statistically significant, and wherein the convex configuration is based on the subset of agents.
 12. The method of claim 7, wherein the host is a robot, and wherein agents of the number of agents are humans.
 13. The method of claim 7, wherein the intent is based on collision avoidance of the host, and the flexibility is based on willingness of the host to deviate from a direct path.
 14. A non-transitory computer readable storage medium storing instructions that when executed by a computer having a processor to perform a method for crowd navigation of a host vehicle among a plurality of agents, the method comprising: identifying a number of agents in a physical environment based on sensor data received from a host; calculating a set of Gaussian processes, wherein the set of Gaussian processes includes a Gaussian Process for each agent of the number of agents; determining an objective function based on an intent and a flexibility for the host and at least two agent of the plurality of agents; and generating a model of the number of agents by applying the objective function to the set of Gaussian processes, wherein the model includes a convex configuration of the physical environment.
 15. The non-transitory computer readable storage medium of claim 14, wherein the sensor data is received from a sensor system of the host.
 16. The non-transitory computer readable storage medium of claim 14, wherein the intent is based on a goal of the host, and wherein the flexibility is based on a willingness of the host to deviate from the goal.
 17. The non-transitory computer readable storage medium of claim 14, wherein the intent is a mean of the objective function and the flexibility is a covariance of the objective function.
 18. The non-transitory computer readable storage medium of claim 14, further comprising: performing optimal shaping to determine a subset of agents of the number of agents that are statistically significant, and wherein the convex configuration is based on the subset of agents.
 19. The non-transitory computer readable storage medium of claim 14, wherein the host is a robot, and wherein agents of the number of agents are humans.
 20. The non-transitory computer readable storage medium of claim 14, wherein the intent is based on collision avoidance of the host, and the flexibility is based on willingness of the host to deviate from a direct path. 